Two Permutation Classes related to the Bubble Sort Operator

Marilena Barnabei, Flavio Bonetti, Matteo Silimbani

Abstract

We introduce the Dual Bubble Sort operator $\hat{B}$ (a sorting algorithm such that, if $\sigma=\alpha\,1\,\beta$ is a permutation, then $\hat{B}(\sigma)=1\,\alpha\,\hat{B} (\beta)$) and consider the set of permutations sorted by the composition $\hat{B}B$, where $B$ is the classical Bubble Sort operator. We show that this set is a permutation class and we determine the generating function of the descent and fixed point distributions over this class. Afterwards, we characterize the same distributions over the set of permutations that are sorted by both $\hat{B}^2$ and $B^2$.